Evaluation of RISK survival models

This document highlights the use of

for the evaluation (RRPlot), and calibration of cox models (CoxRiskCalibration) or logistic models (CalibrationProbPoissonRisk) of survival data.

Furthermore, it can be used to evaluate any Risk index that reruns the probability of a future event on external data-set.

This document will use the survival::rotterdam, and survival::gbsg data-sets to train and predict the risk of cancer recurrence after surgery. Both Cox and Logistic models will be trained and evaluated.

Here are some sample plots returned by the evaluated functions:

The libraries

library(survival)
library(FRESA.CAD)
## Loading required package: Rcpp
## Loading required package: stringr
## Loading required package: miscTools
## Loading required package: Hmisc
## 
## Attaching package: 'Hmisc'
## The following objects are masked from 'package:base':
## 
##     format.pval, units
## Loading required package: pROC
## Type 'citation("pROC")' for a citation.
## 
## Attaching package: 'pROC'
## The following objects are masked from 'package:stats':
## 
##     cov, smooth, var
source("~/GitHub/FRESA.CAD/R/RRPlot.R")
source("~/GitHub/FRESA.CAD/R/PoissonEventRiskCalibration.R")
op <- par(no.readonly = TRUE)
pander::panderOptions('digits', 3)
pander::panderOptions('table.split.table', 400)
pander::panderOptions('keep.trailing.zeros',TRUE)

Breast Cancer Royston-Altman data

survival::rotterdam conditioning

gbsgdata <- gbsg
rownames(gbsgdata) <- gbsgdata$pid
gbsgdata$pid <- NULL

odata <-rotterdam
rownames(odata) <- odata$pid
odata$pid <- NULL
odata$rfstime <- odata$rtime
odata$status <- odata$recur
odata$rtime <- NULL
odata$recur <- NULL

odata <- odata[,colnames(odata) %in% colnames(gbsgdata)]

odata$size <- 10*(odata$size=="<=20") + 
  35*(odata$size=="20-50") + 
  60*(odata$size==">50")

data <- as.data.frame(model.matrix(Surv(rfstime,status)~.*.,odata))

data$`(Intercept)` <- NULL

dataBrestCancerTrain <- cbind(time=odata[rownames(data),"rfstime"],status=odata[rownames(data),"status"],data)

colnames(dataBrestCancerTrain) <-str_replace_all(colnames(dataBrestCancerTrain),":","_")
colnames(dataBrestCancerTrain) <-str_replace_all(colnames(dataBrestCancerTrain)," ","")
colnames(dataBrestCancerTrain) <-str_replace_all(colnames(dataBrestCancerTrain),"\\.","_")
colnames(dataBrestCancerTrain) <-str_replace_all(colnames(dataBrestCancerTrain),"-","_")
colnames(dataBrestCancerTrain) <-str_replace_all(colnames(dataBrestCancerTrain),">","_")
dataBrestCancerTrain$time <- dataBrestCancerTrain$time/365 ## To years


pander::pander(table(odata[rownames(data),"status"]),caption="rotterdam")
rotterdam
0 1
1464 1518

The survival::gbsg data conditioning

gbsgdata <- gbsgdata[,colnames(odata)]
data <- as.data.frame(model.matrix(Surv(rfstime,status)~.*.,gbsgdata))

data$`(Intercept)` <- NULL

dataBrestCancerTest <- cbind(time=gbsgdata[rownames(data),"rfstime"],status=gbsgdata[rownames(data),"status"],data)

colnames(dataBrestCancerTest) <-str_replace_all(colnames(dataBrestCancerTest),":","_")
colnames(dataBrestCancerTest) <-str_replace_all(colnames(dataBrestCancerTest)," ","")
colnames(dataBrestCancerTest) <-str_replace_all(colnames(dataBrestCancerTest),"\\.","_")
colnames(dataBrestCancerTest) <-str_replace_all(colnames(dataBrestCancerTest),"-","_")
colnames(dataBrestCancerTest) <-str_replace_all(colnames(dataBrestCancerTest),">","_")
dataBrestCancerTest$time <- dataBrestCancerTest$time/365

pander::pander(table(odata[rownames(data),"status"]), caption="gbsg")
gbsg
0 1
499 183

Cox Modeling

ml <- BSWiMS.model(Surv(time,status)~.,data=dataBrestCancerTrain,loops=0)
sm <- summary(ml)
pander::pander(sm$coefficients)
  Estimate lower mean upper u.Accuracy r.Accuracy full.Accuracy u.AUC r.AUC full.AUC IDI NRI z.IDI z.NRI Delta.AUC Frequency
size_nodes -8.55e-04 -0.001403 -8.45e-04 -0.000231 0.624 0.644 0.646 0.629 0.645 0.647 3.27e-03 0.33343 7.01 9.3054 -0.00172 1
nodes 1.88e-01 0.082234 1.85e-01 0.294899 0.637 0.648 0.646 0.640 0.649 0.647 3.84e-03 0.13951 5.59 3.8202 0.00260 1
grade 5.39e-01 0.311955 5.38e-01 0.792299 0.565 0.645 0.646 0.561 0.646 0.647 5.54e-03 0.11567 4.97 3.6766 -0.00104 1
age -7.42e-03 -0.012659 -7.12e-03 -0.002620 0.513 0.629 0.646 0.513 0.629 0.647 3.48e-03 0.09715 4.90 2.6559 -0.01764 1
grade_nodes -2.91e-02 -0.055106 -2.89e-02 -0.001119 0.635 0.648 0.646 0.639 0.648 0.647 1.29e-03 -0.08062 3.62 -2.2618 0.00182 1
size 2.41e-02 0.000207 2.42e-02 0.047018 0.595 0.642 0.646 0.595 0.643 0.647 1.60e-03 0.00156 3.41 0.0425 -0.00392 1
size_grade -2.92e-03 -0.011304 -2.97e-03 0.005226 0.598 0.643 0.646 0.599 0.644 0.647 3.72e-04 -0.06105 2.33 -1.6745 -0.00304 1
age_nodes 5.58e-05 -0.000656 1.15e-05 0.001068 0.626 0.645 0.646 0.630 0.645 0.647 5.06e-05 0.06168 2.10 1.6845 -0.00140 1

Cox Model Performance

Here we evaluate the model using the RRPlot() function.

The evaluation of the raw Cox model with RRPlot()

Here we will use the predicted event probability assuming a baseline hazard for events withing 5 years

index <- predict(ml,dataBrestCancerTrain)
timeinterval <- 5 # Five years

h0 <- sum(dataBrestCancerTrain$status & dataBrestCancerTrain$time < timeinterval)
h0 <- h0/nrow(subset(dataBrestCancerTrain,time > timeinterval | status==1))

rdata <- cbind(dataBrestCancerTrain$status,ppoisGzero(index,h0))

rrAnalysisTrain <- RRPlot(rdata,atProb=c(0.90,0.80),
                     timetoEvent=dataBrestCancerTrain$time,
                     title="Raw Train: Breast Cancer",
                     ysurvlim=c(0.00,1.0),
                     riskTimeInterval=timeinterval)

ROC: Raw Train: Breast Cancer

As we can see the Observed probability as well as the Time vs. Events are not calibrated.

Analyzing RRPlot() outputs

pander::pander(t(rrAnalysisTrain$OERatio),caption="O/E Ratio")
O/E Ratio
est lower upper
1.12 1.07 1.18 
pander::pander(t(rrAnalysisTrain$ROCAnalysis$aucs),caption="ROC AUC")
ROC AUC
est lower upper
0.695 0.677 0.714 
pander::pander((rrAnalysisTrain$ROCAnalysis$sensitivity),caption="Sensitivity")
Sensitivity
est lower upper
0.304 0.281 0.328 
pander::pander((rrAnalysisTrain$ROCAnalysis$specificity),caption="Specificity")
Specificity
est lower upper
0.9 0.883 0.915 
pander::pander(rrAnalysisTrain$c.index,caption="C. Index")
C Index Dxy S.D. n missing uncensored Relevant Pairs Concordant Uncertain
0.677 0.354 0.0141 2982 0 1518 6184528 4188057 2703838 
pander::pander(t(rrAnalysisTrain$thr_atP),caption="Probability Thresholds")
Probability Thresholds
90% 80%
0.474 0.397 
pander::pander(t(rrAnalysisTrain$RR_atP),caption="Risk Ratio")
Risk Ratio
est lower upper
1.71 1.6 1.82 
pander::pander(rrAnalysisTrain$sufdif,caption="Logrank test")
Logrank test Chisq = 474.669311 on 2 degrees of freedom, p = 0.000000
  N Observed Expected (O-E)^2/E (O-E)^2/V
class=0 1983 812 1143 96.0 393.7
class=1 390 244 175 27.1 30.7
class=2 609 462 200 344.4 401.7

Cox Calibration

op <- par(no.readonly = TRUE)


calprob <- CoxRiskCalibration(ml,dataBrestCancerTrain,"status","time")

pander::pander(c(h0=calprob$h0,
                 Gain=calprob$hazardGain,
                 DeltaTime=calprob$timeInterval),
               caption="Cox Calibration Parameters")
h0 Gain DeltaTime
0.693 2.32 6.86

The RRplot() of the calibrated model

h0 <- calprob$h0
timeinterval <- calprob$timeInterval;

rdata <- cbind(dataBrestCancerTrain$status,calprob$prob)


rrAnalysisTrain <- RRPlot(rdata,atProb=c(0.90,0.80),
                     timetoEvent=dataBrestCancerTrain$time,
                     title="Train: Breast Cancer",
                     ysurvlim=c(0.00,1.0),
                     riskTimeInterval=timeinterval)

ROC: Train: Breast Cancer

Analyzing RRPlot() outputs of the calibrated model

pander::pander(t(rrAnalysisTrain$OERatio),caption="O/E Ratio")
O/E Ratio
est lower upper
0.992 0.942 1.04 
pander::pander(t(rrAnalysisTrain$ROCAnalysis$aucs),caption="ROC AUC")
ROC AUC
est lower upper
0.695 0.677 0.714 
pander::pander((rrAnalysisTrain$ROCAnalysis$sensitivity),caption="Sensitivity")
Sensitivity
est lower upper
0.304 0.281 0.328 
pander::pander((rrAnalysisTrain$ROCAnalysis$specificity),caption="Specificity")
Specificity
est lower upper
0.9 0.883 0.915 
pander::pander(rrAnalysisTrain$c.index,caption="C. Index")
C Index Dxy S.D. n missing uncensored Relevant Pairs Concordant Uncertain
0.677 0.354 0.0141 2982 0 1518 6184528 4188057 2703838 
pander::pander(t(rrAnalysisTrain$thr_atP),caption="Probability Thresholds")
Probability Thresholds
90% 80%
0.646 0.559 
pander::pander(t(rrAnalysisTrain$RR_atP),caption="Risk Ratio")
Risk Ratio
est lower upper
1.71 1.6 1.82 
pander::pander(rrAnalysisTrain$sufdif,caption="Logrank test")
Logrank test Chisq = 474.669311 on 2 degrees of freedom, p = 0.000000
  N Observed Expected (O-E)^2/E (O-E)^2/V
class=0 1983 812 1143 96.0 393.7
class=1 390 244 175 27.1 30.7
class=2 609 462 200 344.4 401.7

Performance on the external data set

index <- predict(ml,dataBrestCancerTest)
pp <- predictionStats_binary(cbind(dataBrestCancerTest$status,index),plotname="Breast Cancer")

Breast Cancer 

par(op)


prob <- ppoisGzero(index,h0)
rdata <- cbind(dataBrestCancerTest$status,prob)
rrAnalysis <- RRPlot(rdata,atThr=rrAnalysisTrain$thr_atP,
                     timetoEvent=dataBrestCancerTest$time,
                     title="Test: Breast Cancer",
                     ysurvlim=c(0.00,1.0),
                     riskTimeInterval=timeinterval)

ROC: Test: Breast Cancer 

par(op)

The Evaluation results of validation data set

pander::pander(t(rrAnalysis$OERatio),caption="O/E Ratio")
O/E Ratio
est lower upper
1.28 1.14 1.43 
pander::pander(t(rrAnalysis$ROCAnalysis$aucs),caption="ROC AUC")
ROC AUC
est lower upper
0.66 0.619 0.701 
pander::pander((rrAnalysis$ROCAnalysis$sensitivity),caption="Sensitivity")
Sensitivity
est lower upper
0.371 0.316 0.429 
pander::pander((rrAnalysis$ROCAnalysis$specificity),caption="Specificity")
Specificity
est lower upper
0.84 0.799 0.875 
pander::pander(rrAnalysis$c.index,caption="C. Index")
C Index Dxy S.D. n missing uncensored Relevant Pairs Concordant Uncertain
0.663 0.326 0.0311 686 0 299 266144 176497 203702 
pander::pander(t(rrAnalysis$thr_atP),caption="Probability Thresholds")
Probability Thresholds
90% 80%
0.646 0.559 
pander::pander(t(rrAnalysis$RR_atP),caption="Risk Ratio")
Risk Ratio
est lower upper
1.76 1.51 2.07 
pander::pander(rrAnalysis$sufdif,caption="Logrank test")
Logrank test Chisq = 74.963210 on 2 degrees of freedom, p = 0.000000
  N Observed Expected (O-E)^2/E (O-E)^2/V
class=0 433 152 210.6 16.315 55.983
class=1 80 36 33.4 0.203 0.229
class=2 173 111 55.0 57.067 71.183

Calibrating the index on the test data

calprob <- CoxRiskCalibration(ml,dataBrestCancerTest,"status","time")

rdata <- cbind(dataBrestCancerTest$status,calprob$prob)

rrAnalysis <- RRPlot(rdata,atProb=c(0.90,0.80),
                     timetoEvent=dataBrestCancerTest$time,
                     title="Recalibrated Test: Breast Cancer",
                     ysurvlim=c(0.00,1.0),
                     riskTimeInterval=calprob$timeInterval)

ROC: Recalibrated Test: Breast Cancer

The Evaluation results of calibrated model on test data set

pander::pander(t(rrAnalysis$OERatio),caption="O/E Ratio")
O/E Ratio
est lower upper
1.01 0.901 1.13 
pander::pander(t(rrAnalysis$ROCAnalysis$aucs),caption="ROC AUC")
ROC AUC
est lower upper
0.66 0.619 0.701 
pander::pander((rrAnalysis$ROCAnalysis$sensitivity),caption="Sensitivity")
Sensitivity
est lower upper
0.284 0.234 0.339 
pander::pander((rrAnalysis$ROCAnalysis$specificity),caption="Specificity")
Specificity
est lower upper
0.899 0.865 0.927 
pander::pander(rrAnalysis$c.index,caption="C. Index")
C Index Dxy S.D. n missing uncensored Relevant Pairs Concordant Uncertain
0.663 0.326 0.0311 686 0 299 266144 176497 203702 
pander::pander(t(rrAnalysis$thr_atP),caption="Probability Thresholds")
Probability Thresholds
90% 80%
0.61 0.502 
pander::pander(t(rrAnalysis$RR_atP),caption="Risk Ratio")
Risk Ratio
est lower upper
1.8 1.54 2.11 
pander::pander(rrAnalysis$sufdif,caption="Logrank test")
Logrank test Chisq = 80.872386 on 2 degrees of freedom, p = 0.000000
  N Observed Expected (O-E)^2/E (O-E)^2/V
class=0 486 177 234.2 13.95 65.72
class=1 76 37 27.9 2.98 3.31
class=2 124 85 37.0 62.43 72.18

Logistic Model

Here we train a logistic model on the same data set

## Only label subjects that present event withing five years

dataBrestCancerR <- subset(dataBrestCancerTrain, time>=5 | status==1)
dataBrestCancerR$status <- dataBrestCancerR$status * (dataBrestCancerR$time < 5)
dataBrestCancerR$time <- NULL

ml <- BSWiMS.model(status~1,data=dataBrestCancerR,loops=0)

sm <- summary(ml)
pander::pander(sm$coefficients)
  Estimate lower mean upper u.Accuracy r.Accuracy full.Accuracy u.AUC r.AUC full.AUC IDI NRI z.IDI z.NRI Delta.AUC Frequency
grade_nodes 0.061410 0.049466 0.06285 0.078896 0.682 0.637 0.686 0.649 0.624 0.655 0.06580 0.549 13.66 15.65 -0.03109 1
size_grade 0.007168 0.005401 0.00746 0.009073 0.632 0.682 0.686 0.626 0.646 0.655 0.01787 0.294 6.74 7.73 -0.00865 1
nodes_hormon -0.056681 -0.108718 -0.06026 -0.012566 0.587 0.688 0.686 0.526 0.658 0.655 0.00280 0.455 3.44 12.15 0.00285 1
pgr -0.000438 -0.000787 -0.00044 -0.000134 0.571 0.689 0.686 0.500 0.659 0.655 0.00257 0.198 2.64 5.75 0.00412 1

Logistic Model Performance

op <- par(no.readonly = TRUE)

## No-calibrated
rdata <- cbind(dataBrestCancerTrain$status,predict(ml,dataBrestCancerTrain))
rrAnalysisTrain <- RRPlot(rdata,atProb=c(0.90,0.80),
                     timetoEvent=dataBrestCancerTrain$time,
                     title="Logistic Train: Breast Cancer",
                     ysurvlim=c(0.00,1.0),
                     riskTimeInterval=5.0)

ROC: Logistic Train: Breast Cancer 

par(op)

The Evaluaiton of the training

pander::pander(t(rrAnalysisTrain$OERatio),caption="O/E Ratio")
O/E Ratio
est lower upper
0.965 0.917 1.01 
pander::pander(t(rrAnalysisTrain$ROCAnalysis$aucs),caption="ROC AUC")
ROC AUC
est lower upper
0.687 0.668 0.706 
pander::pander((rrAnalysisTrain$ROCAnalysis$sensitivity),caption="Sensitivity")
Sensitivity
est lower upper
0.323 0.3 0.348 
pander::pander((rrAnalysisTrain$ROCAnalysis$specificity),caption="Specificity")
Specificity
est lower upper
0.9 0.883 0.915 
pander::pander(rrAnalysisTrain$c.index,caption="C. Index")
C Index Dxy S.D. n missing uncensored Relevant Pairs Concordant Uncertain
0.676 0.353 0.0141 2982 0 1518 6184528 4183317 2703838 
pander::pander(t(rrAnalysisTrain$thr_atP),caption="Probability Thresholds")
Probability Thresholds
90% 80%
0.551 0.438 
pander::pander(t(rrAnalysisTrain$RR_atP),caption="Risk Ratio")
Risk Ratio
est lower upper
1.76 1.65 1.87 
pander::pander(rrAnalysisTrain$sufdif,caption="Logrank test")
Logrank test Chisq = 544.494088 on 2 degrees of freedom, p = 0.000000
  N Observed Expected (O-E)^2/E (O-E)^2/V
class=0 1971 800 1145 103.8 428.3
class=1 373 227 171 18.1 20.5
class=2 638 491 202 413.1 483.7

Results on the validation set using Logistic model

pre <- predict(ml,dataBrestCancerTest)
rdata <- cbind(dataBrestCancerTest$status,pre)

rrAnalysis <- RRPlot(rdata,atThr=rrAnalysisTrain$thr_atP,
                     timetoEvent=dataBrestCancerTest$time,
                     title="Logistic Test: Breast Cancer",
                     ysurvlim=c(0.00,1.0),
                     riskTimeInterval=5)

ROC: Logistic Test: Breast Cancer 

par(op)

The evaluation of the validation

pander::pander(t(rrAnalysis$OERatio),caption="O/E Ratio")
O/E Ratio
est lower upper
1.22 1.08 1.36 
pander::pander(t(rrAnalysis$ROCAnalysis$aucs),caption="ROC AUC")
ROC AUC
est lower upper
0.677 0.638 0.717 
pander::pander((rrAnalysis$ROCAnalysis$sensitivity),caption="Sensitivity")
Sensitivity
est lower upper
0.331 0.278 0.388 
pander::pander((rrAnalysis$ROCAnalysis$specificity),caption="Specificity")
Specificity
est lower upper
0.884 0.848 0.914 
pander::pander(rrAnalysis$c.index,caption="C. Index")
C Index Dxy S.D. n missing uncensored Relevant Pairs Concordant Uncertain
0.681 0.361 0.0304 686 0 299 266144 181160 203702 
pander::pander(t(rrAnalysis$thr_atP),caption="Probability Thresholds")
Probability Thresholds
90% 80%
0.551 0.438 
pander::pander(t(rrAnalysis$RR_atP),caption="Risk Ratio")
Risk Ratio
est lower upper
1.89 1.62 2.2 
pander::pander(rrAnalysis$sufdif,caption="Logrank test")
Logrank test Chisq = 101.719564 on 2 degrees of freedom, p = 0.000000
  N Observed Expected (O-E)^2/E (O-E)^2/V
class=0 427 147 211.4 19.6 68.2
class=1 115 53 45.9 1.1 1.3
class=2 144 99 41.7 78.9 93.3

Logistic Model Poisson Calibration

riskdata <- cbind(dataBrestCancerTrain$status,predict(ml,dataBrestCancerTrain),dataBrestCancerTrain$time)
calprob <- CalibrationProbPoissonRisk(riskdata)

timeinterval <- calprob$timeInterval;
gain <- calprob$hazardGain

rdata <- cbind(dataBrestCancerTrain$status,calprob$prob)


rrAnalysisTrain <- RRPlot(rdata,atProb=c(0.90,0.80),
                     timetoEvent=dataBrestCancerTrain$time,
                     title="Adj Logistic Train: Breast Cancer",
                     ysurvlim=c(0.00,1.0),
                     riskTimeInterval=timeinterval)

ROC: Adj Logistic Train: Breast Cancer 

par(op)

The Evaluaiton of the calibrated logistic: training

pander::pander(t(rrAnalysisTrain$OERatio),caption="O/E Ratio")
O/E Ratio
est lower upper
0.994 0.945 1.05 
pander::pander(t(rrAnalysisTrain$ROCAnalysis$aucs),caption="ROC AUC")
ROC AUC
est lower upper
0.687 0.668 0.706 
pander::pander((rrAnalysisTrain$ROCAnalysis$sensitivity),caption="Sensitivity")
Sensitivity
est lower upper
0.323 0.3 0.348 
pander::pander((rrAnalysisTrain$ROCAnalysis$specificity),caption="Specificity")
Specificity
est lower upper
0.9 0.883 0.915 
pander::pander(rrAnalysisTrain$c.index,caption="C. Index")
C Index Dxy S.D. n missing uncensored Relevant Pairs Concordant Uncertain
0.676 0.353 0.0141 2982 0 1518 6184528 4183317 2703838 
pander::pander(t(rrAnalysisTrain$thr_atP),caption="Probability Thresholds")
Probability Thresholds
90% 80%
0.649 0.529 
pander::pander(t(rrAnalysisTrain$RR_atP),caption="Risk Ratio")
Risk Ratio
est lower upper
1.76 1.65 1.87 
pander::pander(rrAnalysisTrain$sufdif,caption="Logrank test")
Logrank test Chisq = 544.494088 on 2 degrees of freedom, p = 0.000000
  N Observed Expected (O-E)^2/E (O-E)^2/V
class=0 1971 800 1145 103.8 428.3
class=1 373 227 171 18.1 20.5
class=2 638 491 202 413.1 483.7 
probLog <- predict(ml,dataBrestCancerTest)
aprob <- adjustProb(probLog,gain)

rdata <- cbind(dataBrestCancerTest$status,aprob)
rrAnalysis <- RRPlot(rdata,atThr=rrAnalysisTrain$thr_atP,
                     timetoEvent=dataBrestCancerTest$time,
                     title="Adj Logistic Test: Breast Cancer",
                     ysurvlim=c(0.00,1.0),
                     riskTimeInterval=timeinterval)

ROC: Adj Logistic Test: Breast Cancer

The evaluation of the calibrated validation

pander::pander(t(rrAnalysis$OERatio),caption="O/E Ratio")
O/E Ratio
est lower upper
1.26 1.12 1.41 
pander::pander(t(rrAnalysis$ROCAnalysis$aucs),caption="ROC AUC")
ROC AUC
est lower upper
0.677 0.638 0.717 
pander::pander((rrAnalysis$ROCAnalysis$sensitivity),caption="Sensitivity")
Sensitivity
est lower upper
0.331 0.278 0.388 
pander::pander((rrAnalysis$ROCAnalysis$specificity),caption="Specificity")
Specificity
est lower upper
0.884 0.848 0.914 
pander::pander(rrAnalysis$c.index,caption="C. Index")
C Index Dxy S.D. n missing uncensored Relevant Pairs Concordant Uncertain
0.681 0.361 0.0304 686 0 299 266144 181160 203702 
pander::pander(t(rrAnalysis$thr_atP),caption="Probability Thresholds")
Probability Thresholds
90% 80%
0.649 0.529 
pander::pander(t(rrAnalysis$RR_atP),caption="Risk Ratio")
Risk Ratio
est lower upper
1.89 1.62 2.2 
pander::pander(rrAnalysis$sufdif,caption="Logrank test")
Logrank test Chisq = 101.719564 on 2 degrees of freedom, p = 0.000000
  N Observed Expected (O-E)^2/E (O-E)^2/V
class=0 427 147 211.4 19.6 68.2
class=1 115 53 45.9 1.1 1.3
class=2 144 99 41.7 78.9 93.3